Let $$\Delta PQR$$ be a triangle. Let $$\vec a = \overrightarrow {QR} ,\vec b = \overrightarrow {RP} $$ and $$\overrightarrow c = \overrightarrow {PQ} .$$ If $$\left| {\overrightarrow a } \right| = 12,\,\,\left| {\overrightarrow b } \right| = 4\sqrt 3 ,\,\,\,\overrightarrow b .\overrightarrow c = 24,$$ then which of the following is (are) true?

In $${R^3},$$ let $$L$$ be a straight lines passing through the origin. Suppose that all the points on $$L$$ are at a constant distance from the two planes $${P_1}:x + 2y - z + 1 = 0$$ and $${P_2}:2x - y + z - 1 = 0.$$ Let $$M$$ be the locus of the feet of the perpendiculars drawn from the points on $$L$$ to the plane $${P_1}.$$ Which of the following points lie (s) on $$M$$?

Match the following :

Column I	Column II
(A) In $ \mathbb{R}^2 $, if the magnitude of the projection vector of the vector $ \alpha \hat{i} + \beta \hat{j} $ on $ \sqrt{3}\hat{i} + \hat{j} $ is $ \sqrt{3} $ and if $ \alpha = 2 + \sqrt{3}\beta $, then possible value(s) of $ |\alpha| $ is (are)	$(P)\ 1$
(B) Let $ \alpha $ and $ b $ be real numbers such that the function $ f(x)= \begin{cases} -3\alpha x^2-2, & x<1 \\[4pt] bx+\alpha^2, & x\ge 1 \end{cases} $ is differentiable for all $ x \in \mathbb{R} $. Then possible value(s) of $ \alpha $ is (are)	$(Q)\ 2$
(C) Let $ \omega \ne 1 $ be a complex cube root of unity. If $ (3-3\omega+2\omega^2)^{4n+3} +(2+3\omega-3\omega^2)^{4n+3} +(-3+2\omega+3\omega^2)^{4n+3}=0, $ then possible value(s) of $ n $ is (are)	$(R)\ 3$
(D) Let the harmonic mean of two positive real numbers $ a $ and $ b $ be $ 4 $. If $ q $ is a positive real number such that $ a,\ 5,\ q,\ b $ is an arithmetic progression, then the value(s) of $ |q-a| $ is (are)	$(S)\ 4$
	$(T)\ 5$